<p>What is studied in this article is a generalized coupled higher-order nonlinear Schrödinger system with variable coefficients, which reveals the technical challenges of transmitting two ultrashort pulses at the same time in the inhomogeneous fibers, and plays a particularly critical role in driving further development in the field of telecommunication. By utilizing the Hirota bilinear method, we derive the Hirota bilinear form of the above system and obtain bright-dark alternating one-, two-, three- and even <i>N</i>-soliton solutions step by step, achieving the expected goal. On this basis, the obtained soliton solutions were further studied to explore the physical meanings contained therein. After assigning some suitable values to the unknown parameters, different types of soliton solutions, such as wave type, U-type and so on, are analyzed and discussed graphically. It is also concluded that parameters <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1319_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha _2(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1319_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha _3(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1319_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha _6(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mn>6</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> jointly affect the overall trajectories of soliton solutions, while the selection of other parameters affects part of the trajectories during the movement. Furthermore, the dynamic behaviors of the two-soliton solutions before and after the collision are studied by means of asymptotic analysis. They are presented intuitively in the form of numerical values such as amplitude, velocity and initial phase. Ultimately, we further study the bilinear Bäcklund transformation of the above system and obtain new soliton solutions. The work done in this paper contributes to a deeper physical understanding of the above system, and provides certain theoretical guidance for further improvements in modern communication, such as increasing transmission capacity, enhancing signal stability and reducing energy consumption.</p>

Discussion on Soliton Solutions and Bilinear Bäcklund Transformation of a Generalized Coupled Higher-Order Nonlinear Schrödinger System with Variable Coefficients

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发表日期：

2025年6月20日

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Yating Hao,
Ben Gao

摘要

What is studied in this article is a generalized coupled higher-order nonlinear Schrödinger system with variable coefficients, which reveals the technical challenges of transmitting two ultrashort pulses at the same time in the inhomogeneous fibers, and plays a particularly critical role in driving further development in the field of telecommunication. By utilizing the Hirota bilinear method, we derive the Hirota bilinear form of the above system and obtain bright-dark alternating one-, two-, three- and even N-soliton solutions step by step, achieving the expected goal. On this basis, the obtained soliton solutions were further studied to explore the physical meanings contained therein. After assigning some suitable values to the unknown parameters, different types of soliton solutions, such as wave type, U-type and so on, are analyzed and discussed graphically. It is also concluded that parameters $\alpha _2(z)$ $α_{2} (z)$ , $\alpha _3(z)$ $α_{3} (z)$ and $\alpha _6(z)$ $α_{6} (z)$ jointly affect the overall trajectories of soliton solutions, while the selection of other parameters affects part of the trajectories during the movement. Furthermore, the dynamic behaviors of the two-soliton solutions before and after the collision are studied by means of asymptotic analysis. They are presented intuitively in the form of numerical values such as amplitude, velocity and initial phase. Ultimately, we further study the bilinear Bäcklund transformation of the above system and obtain new soliton solutions. The work done in this paper contributes to a deeper physical understanding of the above system, and provides certain theoretical guidance for further improvements in modern communication, such as increasing transmission capacity, enhancing signal stability and reducing energy consumption.